Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability
Polytopic Matrix Factorization: Determinant Maximization Based Criterion and Identifiability
复制标题
多面体矩阵分解：基于行列式最大化的标准和可识别性
DOI：
10.1109/tsp.2021.3112918
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发表时间：
2022
影响因子：
5.4
通讯作者：
A. Erdogan
中科院分区：
文献类型：
--
作者：
Gokcan Tatli;A. Erdogan
我们引入了多主题矩阵分解(PMF)作为一种新的数据分解方法。在这个新的框架中，我们将输入数据建模为从多面体提取的一些潜在向量的未知线性变换。在这个意义上，本文考虑了一种半结构化数据模型，其中输入矩阵被建模为一个满列秩矩阵和一个包含多个多面体样本的矩阵的乘积。多面体的选择反映了潜在成分的假定特征及其相互关系。作为因式分解准则，我们提出了潜在向量的样本自相关矩阵的行列式最大化(Det-Max)。我们引入了一个可辨识性的充分条件，它要求隐向量的凸壳包含具有特定紧性约束的多面体的最大体积内接椭球。基于Det-Max准则和所提出的可辨识性条件，我们证明了所有满足特定对称限制的多面体都符合PMF框架。具有无限多个多面体选择为表征潜在向量提供了一种形式的灵活性。具体地说，可以定义具有异质特征的潜在向量，从而能够在子向量级分配诸如非负性和稀疏性的属性。本文提供的例子说明了多面体选择和相应的要素表示之间的联系。
We introduce Polytopic Matrix Factorization (PMF) as a novel data decomposition approach. In this new framework, we model input data as unknown linear transformations of some latent vectors drawn from a polytope. In this sense, the article considers a semi-structured data model, in which the input matrix is modeled as the product of a full column rank matrix and a matrix containing samples from a polytope as its column vectors. The choice of polytope reflects the presumed features of the latent components and their mutual relationships. As the factorization criterion, we propose the determinant maximization (Det-Max) for the sample autocorrelation matrix of the latent vectors. We introduce a sufficient condition for identifiability, which requires that the convex hull of the latent vectors contains the maximum volume inscribed ellipsoid of the polytope with a particular tightness constraint. Based on the Det-Max criterion and the proposed identifiability condition, we show that all polytopes that satisfy a particular symmetry restriction qualify for the PMF framework. Having infinitely many polytope choices provides a form of flexibility in characterizing latent vectors. In particular, it is possible to define latent vectors with heterogeneous features, enabling the assignment of attributes such as nonnegativity and sparsity at the subvector level. The article offers examples illustrating the connection between polytope choices and the corresponding feature representations.